Singular integral equations have found wide use in investigating the ... equations of certain two-dimensional problems have also been constructed for the ..... Let us consider the first basic problem when the boundary of the area L is free of load.
METHOD OF SINGULAR INTEGRAL EQUATIONS PROBLEMS
IN TWO-DIMENSIONAL
STATIC
OF THE THEORY OF CRACKS
M. P. Savruk
UDC 539.375
Singular integral equations have found wide use in investigating the stressed conditic t near straight cracks in two-dimensional areas of various form [i]. The singular integral equations of certain two-dimensional problems have also been constructed for the general case of a system of curved notches in an infinite plane [2-10]. In [Ii] the basic plane problems of the theory of elasticity for a finite or infinite, multiple connected area weakened by cracks and holes of arbitrary form were considered. In this work such generalizations are obtained for certain other boundary problems of the statics of elastic bodies with cracks. Plane Problems of the Theory of Elasticity. Let the area S, occupied by the body, be bound by one or several closed contours LI, L2 . . . . , LM, Lo, where the first M contours are located outside one another and the last envelops all of the others (the contours L k are designated as S~ (k = I, M--) and the infinite area, the exterior of the contour Lo, as So. The positive direction of the circuit of the contours L k (k = i, M) and Lo will be assumed to be that in which S remains to the left. In addition, in the area S there are N--M curved notches L k (k = M + i, N) (Fig. i). Let L = L' + L ' ~ e r e L' and L" designate the aggregates of closed and open contours, i.e., L = ULk(k = 0, N), L' = UL k (k = 0, M), and L" = UL k (k = M+I, N). Let us consider the stresses
the first basic problem,
N+iT=p*(t),
when on the boundary
tEL'; N' + i T ~:= p * ( t )
•
of the area L are specified
t~A".
(1)
Here N and T are the normal and tangential components of the external forces and the superscript "+" ("-") designates the limiting values of the function in approaching the notch from the left (right). At the same time, there exist the relationships
;p*(t)dt + 2~ q(t)dt=O, Re[.fip*(t)dt+2~Tq(t)dJ=O, L"
L~
L'
(2)
L"
which express the equality to zero of the main vector and the main moment of the external forces acting on the contour L. In the case of the second basic problem we will assume specified the derivatives of the displacements
2G--d(u+iv) =f,'(/),
dt
t~L'; 2 0 d (u •
that on the boundary
+-) =[,'(t)--g
2--~/g'(t); 1+
L there are
(3)
tEL" , w h e r e G i s t h e s h e a r m o d u l u s ; u and v a r e t h e c o m p o n e n t s o f t h e d i s p l a c e m e n t s ; • = 3 -- 4p f o r plane strain a n d • = (3 -- p ) / ( 1 + V) f o r t h e p l a n e s t r e s s e d condition; and p i s P o i s s o n ' s ratio~ A l s o known a r e t h e m a i n v e c t o r s o f t h e f o r c e s a p p l i e d t o e a c h c o n t o u r Lk ( k = 0 , N) with the projections Xk a n d Yk on t h e Ox a n d Oy a x e s . Then t h e c o m p l e x p o t e n t i a l s of the stresses r a n d ~ , ( z ) i n t h e a r e a S may b e r e p r e s e n t e d i n t h e f o r m [12]
9 , (z) =
1 ~-~ Xk + iYk + ~ (z). 2~(1-+-~)~_1 z - - z k
G. V. Karpenko Physicomechanical Institute, Academy of Sciences of the Ukrainia~ SSR, L'vov. Translated from Fiziko-Fd~imicheskaya Mekhanika Materialov, Vol. 17, No. 5, pp. 5158, September-October, 1981. Original article submitted February 20, 1981.
0038-5565/81/1705-0429507.50
9 1982 Plenum Publishing
Corporation
429
Fig. i M
x
~ Xk-- iYk ~W(z),
~'*(z)=2,,O+~)
(4)
_~ z - z ~
where the functions ~(z) and ~(z) are holomorphic in the area S, and zk are points arbitrarily located within the contours L k (k = i, M). We will search for the solution of the boundary problems of (i) and (2) in the form [ii]
r
- 1 ~ Q (tt-)z dt , Q(t) = g'(t) - -1-+q2i~
(t),
(5)
L
where
tv (z) = ~
(z - z~)~ + 2=L2 [
- -- z)~ dt I , (t
t -. z
Mk = --2Im~Tg'(t)dt, k = 1, M; tl Lk
and with /EL', q (t) = 0, or g'(t) = 0, respectively, in the case of the first or second basic problem. The functions (5) differ from similar presentations of the complex potentials for a system of notches in an infinite plane [4] only in the presence in the potential ~(z) of terms with M k. It would be possible to use accurately the same representations of ~(z) and ~(z) for closed contours as in the case of notches but then it would be necessary to change the form of Eq~ (4) in such a manner that for the functions O(z) and ~(z) there would be obtained a boundary problem with forces, the main vector and main moment of which are equal to zero on each contour L k individually, specified on the contours L k (k = i, M), Having substituted the potentials (4) and (5) in the boundary conditions the singular integral equation of the first basic problem
2~1f{2Q(t)+t - ~iq(t)dt +k, (t, ~)[Q(t)+2iq (t)]dt + k~(t,r
(i), we obtain
(t) clt}--
L
1 ~ Ak~Mk d~ ~ a d s 2~zi (z--zk) 2 ~ + ~n n~=P(~), ~Ln, n=O,N
(6)
k=O
for determining the unknown function g'(t). Here 6n = 1 with n = 0, M; ~n = 0 with n = M + i, N; Akn = 1 + (~n -- l)~ko; 6kn is the Kronecker symbol, s i s the arc abscissa corre- , sponding to the point T, z 0 = 0 ~ S ; kx (t,x) _--__dIn _t-- -,x' k2 (t,~) ----___d _t --_.~.and d~ t- 9 d~ t --
P (~) = P* (~) + 2~ (1 + ~) ~
(-- k)
~-~k
- z~
J~,
To the left portion of Eq. (6) are added the functionals
430
Mo=--2lm ~g'(t)d--/, a.=~g'(t)dt L"
(n = o, M).
(.7)
Ln
Then Eq. (6) for any right portion has a single solution with fulfillment of the additional conditions
~ g'(t)dt=O,
k=M+I,N,
(8)
resulting from the single-valuedness of the displacements in bypassing the contours of the cracks. With this g'(t) we search in the class of H* functions existing at the ends of the notches fof the integrated singularities and in each closed portion of the line L not containing ends for those satisfying the HUlder conditions~ With fulfillment of the conditions of equilibrium (2) the functionals (7) are equal to zero since the Eq. (6) obtained provides the solution of the problem. From the boundary conditions (3) with the use of the potentials at the singular integral equation of the second basic problem 1
S{(~-I)Q(t)-2iq(t) dt-kl(t,~)[Q(t)+2iq(t)]dl--k~(t,~)Q(f)d[} t--z
(4) and (5) we arrive
r (~), ~ ~Ln, n O + a . b. _.ds= ~ = ,
L
N (9)
relative to the unknown function q(t).
t' (~)
=
h' O) +
2~
1
(1 +
Here
X~--iYa "~ ~[ (z--zk) 2 + ~) ~_ , u(X~+2Y~) .~ - z~ ~ - z~
"~--za
J]
To the left portion of Eq. (9) are added the functionals O
b.=
~ q(Oat,
n=O,m,
(lO)
,i
Ln
which together with the conditions
q(t)dt=~(X.+iY.),
n=M+I,N
La provides the existence of a single solution with any right portion while q(t)~H*. In view of the single-valuedness of the displacements in the areaof S the functionals (i0) are equal to zero so that Eq. (9) determines the solution of the problem. Equations (6) and (9) were constructed in generalization of the integral equations for open contours (notches) to the case of closed. At the same time, the stressed and strained state in the area of S is analytically extended to the areas ST (k = I, M) and St, i.e., to the whole plane in such a manner that in passing through the closed contours L k (k = 0 , M) either the stresses remain continuous and the vector of displacement obtains a jump g(t) (first basic problem) or the displacements remain continuous, while the stresses have a break q(t) (second basic problem). In the absence of closed contours Lk (k = 0, M) Eqs. (6) and (9) coincide with the integral equations of the basic boundary problems in the case of a system of curved notches in an infinite plane [4, 5]. Consequently, in the solution of the plane problem of the theory of elasticity for multiply connected areas containing both holes and cracks of arbitrary form in [ii] a single approach was propose d . Below this will be extended to other two-dimensional problems for similar areas. Antiplane Problem of the Theory of Elasticity. Let a cylindrical body, the cross section of which has the form of the area S (Fig. i), be under conditions of antiplane deformation and the axis of deformation be directed along the z axis of the Cartesian system of coordinates (x, y, z). Then the stressed and strained state of the body is determined through the single harmonic function of displacement w(x, y), which may be represented in the form
Ow(x, V)=Ref.(z), z=x+ig. 431
where f,(z)
is an analytical
Let u s consider stresses
function in the area So
the first basic problem,
oow=~,(t), t~L';
o a~*-= o, (t) +- ~ (t), t ~ L",
On
satisfying
the condition
when on the contour L there are specified
the
(11)
On
of equilibrium
.I *, (Jds + 2 ~ l~(z)ds----O. L'
In Eq s. (ii) n designates contour n".
(12)
L'~
the external
normal
to the contour L' or to the left edge of the
Let Z k be the main vector of the forces applied to the contour L k (k = 0, N). complex potential F,(z) = f~(z) may be represented in the form
1
F, (z)
Then the
~ -z,,~----7+ F (z),
(13)
k-I
where the function F(z) is holomorphic in the area S. Let us make an analytical extension from the area S into the areas S~ (k = i, M) and So so that in passing through the closed contour L' the stresses remained continuous and the displacements made the jump y(t). Then the potential F(z) is written in the form [8]
~(z) and in the case considered
1 [/4
=
ds (:) d~ H ( 0 = ~' O) + i~ (~) d~'
~Tj :,-L
(1/4)
7 '
~(T) = 0 with ~ L ' .
In satisfying the boundary the first basic problem
conditions
[, t - - ~ --[--kl ( [ ' Z ) [ 2 r t
(Ii) we obtain
(t) --
the singular
d~
d~'
integral equations
rt = 0 , N
Of
(15)
L
for determining
the unknown
Here
function y'(t).
M
(,j=~,(~)_ 1%-~Zklm To the left portion
of Eq.
(15) are added
( 1
d'~
"dss)"
the functionals
on: ['("(t)d4 n=O,A/! ,3 L lz
equal to zero with al conditions
fulfillment
of the conditions
~
of equilibrium
(12). Then with
the addition-
n=M+],N
'F(t)clt=o,
Ln
Eq.
(15) h a s i n t h e c l a s s
of f u n c t i o n s
o f H* a s i n g l e
solution
In the case of the second basic problem we a~sume fied the derivatives from the displacements
o d_w = ,-I,,' (0, t ~ L'; ~ aw '- = r dt
w i t h any f u n c t i o n
o(~).
that on the contour L there are speci-
(t) + ~' (t),
t ~ L",
( 16 )
dt
and the main vectors Z k (k = 0," N) of the external forces are also known. The complex potential F~(z) is represented in the form (13) and (14) where 7'(~) = 0 w i t h ~ L ' , i.e., we add the area S to the whole plane in such a manner that in passing through the closed contour
432
L ~ the displacements were continuous and the stresses made the jump ~(T). From the boundary conditions (16) we arrive at the singular integral eGuation of the second basic oroblem
1 ~{ 2 H (t)t-~----.;' (t) +k~(t,~)[H(t)-2~'(t)] } dt+~ n acts= n~ ~'('~),~L,,,r~=0, N ~-~
(17)
L
relative to the unknown jump in stresses ~(t).
*' (~) ~ r To t h e
left
portion
o f Eq.
(17)
are
Here
(~) - 2G, d-~ k = l introduced
G =~(t)ds,
the
~--~ functionals
(18)
n=o,M,
Ln
which together with the conditions
f ~(t)ds=Z., n----M+ I,N Ln
provides the existence of a single solution ~(t)~H, of Eq. (17) with any right portion. In view of the single-valuedness of the displacements the functionals (18) are equal to zero so that Eq. (17) solves the problem. Plane Problem of Thermoelasticity. In solving the plane steady problem of thermal conductivity it is necessary to find the temperature distribution in the area S described by the harmonic function T(x, y) with certain conditions on the boundary of the area L. In the case of first or second order boundary conditions, i.e.,
T=9.(t),
t~L'; Tz~-%(t) •
t~L"
(19)
OT
t~L';3T +- = o . ( t )
t~L",
(20)
or
--=
~.(t),
+- ~ ( 0 ,
problems are similar to the above-considered basic boundary antiplane problems of the theory of elasticity. these
Let us dwell on the solution of problems of thermoelasticity when on the boundary L there are specified the conditions (20) with ~(t) = 0. In particular as a result with ~ ( t ) = 0 (t~L") the important practical case of thermally isolated cracks is obtained. Then the temperature T(x, y) is determined by equations T (x y ) = Re
:,
:, (z) = --'2~-'-X I I J - -(t) q~ln(z--zJ+ t(z), f(z) =--,'rrt , dt k=l t -- Z
(21)
L where X is the coefficient of thermal conductivity, qk i s t h e t o t a l h e a t f l o w o n t h e c o n t o u r Lk (k = 1 , M) d i r e c t e d i n t o t h e a r e a S, a n d t h e f u n c t i o n y ' ( t ) i s t h e s o l u t i o n o f Eq. ( 1 5 ) with D(t) = 0.
Let us consider the first basic problem when the boundary of the area L is free of load and the crack edges are not in contact, The complex potentials of the stresses ~,(z) and ~:=(z) are represented in the form [12, 13] 1
M
a,, (z) = G ~
m~ [in (z -- z~) + 11 § ~ (z),
k--1
~ * ( z ) = - 2 C ~k = I Lz-z~ (z-zk)q +r w h e r e ~ ( z ) a n d ~ ( z ) ' a r e h o l o m o r D h i c f u n c t i o n s i n t h e a r e a S- Mk i s t h e m a i n moment o f e x t e r n a l f o r c e s a c t i n g on t h e c o n t o u r Lk ( k = i , - M ) ; mk = ~ q k / ( t ( l + z ) ) ; ~ = mE f o r p l a n e s t r a i n ; B = a E / ( 1 + ~) f o r t h e p l a n e s t r e s s e d condition; E is Young's modulus; and a is the temperat u r e coefficient of linear expansion.
433
Let us find the solution of the auxiliary problem when on the whole plane in passing through the contour L the stresses corresponding to the potentials ~(z), ~(:z), and f(z) are continuous and the vector of displacements makes the jump g(t). Having utilized the equation of the components of the stresses ox, oy, zxy and displacements u and v through the complex potentials [12, 13]
~x -F zy = 2 [ ~ (z) -'F ~ (z)],
~y - ox -t- 2i~y = 2 [z{b' (z) + W (z)], "
2O (u + iv) = x~(z) -- z@ (z) -- + (z) + ~ I f (z) dz' )
we obtain the same boundary problem
'~+ (0 - ~ - (0 = i [g' (t) + 2 i ~ (0/(1 + ~')l - - i O (t), ~/~ dF, (t)l+--[ToY(t)+~F(t)l-=i[O(t)--O()]~t t~L,
[t-O'(t) + 9 as in the above-considered
case of the force problem. As a result, we find
L
L
lt 7
Satisfying w i t h t h e u s e o f E q s . (22) and ( 2 3 ) t h e b o u n d a r y c o n d i t i o n s (1) with zero right portions, w e arrive at the singular integral equation of the first basic problem of the rmo elas ti city 1~{[ 2~
2 q_kx(t,~)]O(t)dt_bk2(t,,)O(t)dt} t-- z
-
L
ds 1 ~ Ak.M k d~2~i ~ (~--z~) - 2 ~ + ~" a n - = s ( ~ ) ' d'~
k=0
relative tothe function g'(t) or G(t) o
s(J=
----~
(24)
x~Ln, n = O , N
Here
2~ =1
m k 21n~
--zk[-F2+
~-z~
and the remaining designations are the same as in Eq. (6)9 At the same time, the zero functionals are added to the left portion of Eq. (24), which together with conditions (8) provide the existence of a single solution of g'(1)~II*. In comparing Eqs. (6) and (24) we may see the analogy between the first basic problems of the theory of elasticity and thermoelasticity. Such an analogy occurs in the case of the second basic problem. Let the body be located in a temperature field (21) and on the bounddry L there are specified the conditions (3). Then it may be shown that the integral equation of the problem of thermoelasticity coincides with Eq. (9) with the only difference that to Q(t) must be added 2i~y(1)/(l+• and to f'(T)
i
mk
Problems of thermoelasticity when t h e considered in a similar manner,
9
temperature
(19)is
specified
on t h e b o u n d a r y
L may b e
The i n t e g r a l presentations of the complex potentials and t h e i n t e g r a l equations remain c o r r e c t f o r an i n f i n i t e p l a n e w e a k e n e d b y h o l e s and n o t c h e s w h e n t h e c o n t o u r Lo i s a b s e n t . At t h e same t i m e , i t i s o b v i o u s t h a t i n E q s . ( 6 ) and ( 2 4 ) No = 0 m u s t b e a s s u m e d and i t i s also necessary t o a c c e p t t h e c o n d i t i o n qx + q2 + . . . + qM = 0 p r o v i d i n g b o u n d e d n e s s i n t h e infinity of the temperature stresses~ We should note that the integral Eqs. (6). and (9) have been given in a somewhat different form in [ii], where each contour L k (k = 0, N) is linked with a local system of Cartesian coordinates xkOkYk. The use of local coordinates is convenient in solving integral equations and also in taking into consideration the different nature of symmetry and periodicity of the problem.
434
The above-used approach to ply connected areas weakened by particularly to problems on the [9, i0] corresponding solutions
the solution of certain two-dimensional problems for multiholes and notches may be extended to other boundary problems, bending of plates and g~mtly s~oping shells, since there are for a system of curved notches in an infinite plane~ LITERATURE CITED
i. 2.
3.
4.
5. 6. 7. 8. 9. i0. ii. 12. 13.
V. V. Panasyuk, M. P. Savruk, and A. P. Datsyshin, The Stress Distribution near Cracks in Plates and Shells [in Russian], Naukova Dumka, Kiev (1976). Ao M. Lin'kov, "Integral equations of the theory of elasticity for a plane with notches leaded with counterbalanced systems of forces," Dokl. Akad. Nauk SSSR, 218, No. 6, 12941297 (1974) o L. A. Fil'shtinski, "The elastic equilibrium of a plane anisotropic medium weakened by arbitrary curved cracks. The limiting transition to an isotropic medium," Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 5, 91-97 (1976). M. P. Savruk, "Contruction of integral equations for two-dimensional problems of the theory of elasticity for a body with curved cracks," Fiz.-Khim. Mekh. Mater~, No. 6, 111-113 (1976). M. P. Savruk, "A system of curved cracks in an elastic body with different boundary conditions on their edges," Fiz.-Khim. Mekh. Mater., No. 6, 74-84 (1978). L. A. Fil'shtinskii, "longitudinal shear in an isotropic medium with notches," Izv. Akad~ Nauk SSSR, Mekh. Tverd. Tela, No. 4, 68-72 (1978). G. S. Kit and M. G. Krivtsun, "The composite problem of thermoelasticity for a plane with curved notches," Dop. Akad~ Nauk UkrSSR, Ser. A, No. 3, 227-230 (1978). M. P. Savruk, "A system of curved notches in an elastic body in antiplane strain," Fiz.Khim. Mekh. Mater., No. 4, 92-98 (1979). M. P. Savruk, "Bending of thin elastic plates weakened by cracks," Fizo-Khim. Mekh. Mater., No. 4, 78-84 (1980). M. P. Savruk~ "Basic boundary problems of the statics of elastic gently-sloping shells with curved cracks," Fiz.-Khim. Mekh. Mater.; No. 3, 50-59 (1981). M. P. Savruk, "Plane problems of the theory of elasticity for a multiply connected area with holes and cracks," Fiz.-Khim. Mekh. Mater~ No. 5, 51-56 (1980). N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity fin Russian], Nauka, Moscow (1966). I. A. Prusov, Some Problemsof Thermoelasticity [in Russian], Izd. Belorus. Univ., Minsk (1972).
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